Improving the accuracy of the results presented here would require at least a thorough study of the inelastic \(N\!N\pi \)  channel, but at the same time a more systematic connection to the Cottingham formula would likely become necessary as well: while the identification of the leading elastic effects is rather intuitive, the extension to subleading corrections is not.
Additional experimental input could, in principle, separate \({\cal C}_1\)  from \({\cal C}_2\) , e.g., via CIB in nuclei, but such an extraction also requires the development of a suitable theoretical framework.
While our results allow for first phenomenological estimates of the impact of the contact term on \(0\nu \beta \beta \)  decay rates, they thus also define a benchmark for future lattice-QCD calculations [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}. In addition to comparing the final result for \({\cal A}_\nu \) , there could also be aspects of the matching strategy and spectral representation, as described in Section , that might prove synergistic between the two approaches. Finally, having concentrated on the LO contact term for light Majorana exchange in this paper, we remark that contact-term contributions arise at LO for other operators mediating \(0\nu \beta \beta \)  decay as well, both at dimension 7 and dimension 9 [8]}, or through exchange of massive sterile neutrinos [9]}, and generalizations of the strategies presented here could help constrain the contact terms in these mechanisms.
